Problem: Let $f(x) = -5x^{2}-6x+6$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )?
Solution: The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-5x^{2}-6x+6 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -5, b = -6, c = 6$ $ x = \dfrac{+ 6 \pm \sqrt{(-6)^{2} - 4 \cdot -5 \cdot 6}}{2 \cdot -5}$ $ x = \dfrac{6 \pm \sqrt{156}}{-10}$ $ x = \dfrac{6 \pm 2\sqrt{39}}{-10}$ $x =\dfrac{3 \pm \sqrt{39}}{-5}$